1. Pseudorandom generators with optimal seed length for non-boolean poly-size circuits Sergei Artemenko Ronen Shaltiel University of Haifa

2. Goal: Reduce randomness complexity of sampling procedures [DI06]. C UnUn P C’ UrUr ≈P n r

3. non-boolean PRGs nb-PRGs [DI06] C pseudorandom ≈P UrUr G C’

4. Our results: nb-PRGs with optimal seed length r=O(l) What type of circuits ?

5. Our results: Formal statement

6. Unconditionl nb-PRGs for poly- size constant-depth circuits

7. Proof outline for main theorem Hardness assumption PRG against conditional tests nb-PRG

8. PRGs for conditional tests {x:A(x)=1} {0,1} n

9. Proof outline for main theorem Hardness assumption Boolean PRG for poly-size nondeterministic NP-circuits PRG against conditional tests nb-PRG [SU05,NW88,KvM99] Previous slide Approach follows [SU05]

10. Intuition for PRGS for conditional tests (following [SU05]) {x:A(x)=1} {0,1} n h’:“1-1 on A” y=G’ (s 2 ) x {0,1} n w h -1 Random hash function h costs >n random bits. Can choose h=h G1(s1) where G 1 is a PRG for nondet NP-circuits.

11. Conclusion and Open Problems Open Problems: More applications of nb-PRGs/cd-PRGs? Replace nondeterministic NP-circuits with nondeterministic circuits. Actually, is hardness for nondeterministic circuits implied by cd-PRGs/nb-PRGs?

12. Thank You…